Table of Contents
Proposition 1 Theorem 1
The forces that keeps the planets in their orbits comes from Jupiter’s center.
The same thing affects those that are affected by Saturn.
The former part of this Proposition appears from Pham. I, and Prop. to Jupiter s centre
Proposition 2 Theorem 2
The primary planets are retained in their orbits by the sun’s force. This force depends on the distance of those planets from the sun
The former part of the Proposition is manifest from Phæn. V, and Prop. II, Book I.
The latter from Phæn. IV, and Cor. 6, Prop. IV, of the same Book.
But this part of the Proposition is, with great accuracy, demonstrable from the quiescence of the aphelion points; for a very small aberration from the reciprocal duplicate proportion would (by Cor. 1, Prop. XLV, Book I) produce a motion of the apsides sensible enough in every single revolution, and in many of them enormously great.
Proposition 3 Theorem 3
The moon is retained in its orbit by the force from the Earth’s center which is reciprocally as the square of the distance of its place from the earth’s centre.
The former part of the Proposition is evident from Phæn. 6, and Prop. 2 or 3, Book 1.
The latter from the very slow motion of the moon’s apogee; which in every single revolution amounting but to 3° 3′ in consequentia, may be neglected.
For (by Cor. 1. Prop. XLV, Book I) it appears, that, if the distance of the moon from the earth’s centre is to the semi-diameter of the earth as D to 1, the force, from which such a motion will result, is reciprocally as D²4⁄243, i. e., reciprocally as the power of D, whose exponent is 24⁄243; that is to say, in the proportion of the distance something greater than reciprocally duplicate, but which comes 59¾ times nearer to the duplicate than to the triplicate proportion.
But in regard that this motion is owing to the action of the sun (as we shall afterwards shew), it is here to be neglected.
The action of the sun, attracting the moon from the earth, is nearly as the moon’s distance from the earth; and therefore (by what we have shewed in Cor. 2, Prop. XLV, Book I) is to the centripetal force of the moon as 2 to 357,45, or nearly so; that is, as 1 to 17829⁄40.
If we neglect so inconsiderable a force of the sun, the remaining force, by which the moon is retained in its orb, will be reciprocally as D². This will yet more fully appear from comparing this force with the force of gravity, as is done in the next Proposition.
Corollary
If we augment the mean centripetal force by which the moon is retained in its orb, first in the proportion of 17729⁄40 to 17829⁄40, and then in the duplicate proportion of the semi-diameter of the earth to the mean distance of the centres of the moon and earth, we shall have the centripetal force of the moon at the surface of the earth; supposing this force, in descending to the earth’s surface, continually to increase in the reciprocal duplicate proportion of the height.
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